Here's what the spec of the AD712 says regarding operating either input close to the negative power rail. I say this because you haven't shown the power rails and, like @Rennex, I suspect you are not using a negative rail.
Typically exceeding −14.1 V negative common-mode voltage on either
input results in an output phase reversal.
This was taken from the data sheet of the device when connected to a +/-15V supply and if your negative rail is in fact 0V, then going to within 0.9 volts of 0V will cause an inversion in the op-amp which will likely result in positive feedback and hence oscillation as you observe.
Lifting V1 up a few volts will likely stop that inversion and the op-amp stops oscillating. So if V1 is at 10V, then V+ is about 1 volt and it's going to be round about the area where input inversion stops taking place.
OK, it's not a long shot but, in my water, I feel this is the problem.
The other issue is that you appear to be saying that your scope probe is connected to V-. Try connecting it to 0V for sensible results.